The model used in this work is built upon the brain-inspired replay (BI-R) model proposed by van de Ven et al [1]}. This combines a softmax classifier with a VAE to generate samples based on previously seen classes and replay them alongside the current inputs, as depicted in Fig. REF . The aim of a VAE's encoder is to distil the high-dimensional input space, \(\mathbf {x} \in \mathbb {R}^{n}\) , into a lower-dimensional latent space, \(\mathbf {z} \in \mathbb {R}^{m}\)  (where \(m<n\) ), by approximating the intractable posterior, \(p(\mathbf {z}|\mathbf {x})\) , as a multivariate Gaussian distribution, \(q_{\mathbf {\phi }}(\mathbf {z}|\mathbf {x})\) , via variational inference. This Gaussian is defined via its mean vector, \(\mathbf {\mu }\) , and the logarithm of its variance, \(\log \mathbf {\sigma ^2}\) , which are returned via a split layer after the encoder and two FC layers, as depicted in Fig. REF . The latent variables sampled from this distribution are then passed through two more FC layers and the decoder, which acts as the conditional distribution, \(p_{\mathbf {\theta }}(\mathbf {x^{\prime }}|\mathbf {z})\) , returning reconstructed samples, \(\mathbf {x^{\prime }}\) . The VAE is then optimised over successive batches by minimising the per-batch loss function:
\(\begin{aligned}&L(\mathbf {x},\mathbf {x}^{\prime },\mathbf {z};\mathbf {\phi },\mathbf {\theta },\mathbf {\lambda }) = \\&\sum ^{N_{s}}_{i=1}\left[ \lambda _{r}\ell _{rec}(\mathbf {x}_i,\mathbf {x}^{\prime }_i;\mathbf {\phi },\mathbf {\theta })\right.\left.+ \lambda _{d}\ell _{reg}(\mathbf {x}_i;\mathbf {\phi },\mathbf {\lambda }) \right], \end{aligned}\) 
